Local sharp maximal functions
type: Item
instance of: scholarly article
title: Local sharp maximal functions
zbMATH Open document ID: 0565.42009
DOI: 10.1016/0021-9045(85)90102-9
published in: Journal of Approximation Theory
publication date: 1985-01-01T00:00:00Z
review text: The authors consider in detail the propserties of the local maximal function introduced by F. John namely, for \(0<\alpha \leq\) set \[ M^{\#}_{0,\alpha}f(x)=\sup_{x\in Q}\inf_{c} \inf \{A\geq 0: | \{y\in Q: | \quad f(y)-c| <A\}| <\alpha | Q| \}, \] where \(c\in {\mathbb{C}}\) and Q denotes a cube in \({\mathbb{R}}^ n\) of sides parallel to the coordinate axes. Among other things they show that the Peetre K-functional for real interpolation between the Lorentz space \(L^{1,\infty}({\mathbb{R}}^ n)\) and BMO(\({\mathbb{R}}^ n)\) satisfies \[ K(t,f;L^{1,\infty}({\mathbb{R}}^ n),BMO({\mathbb{R}}^ n))\approx \sup_{0<s<t}s(M^{\#}_{0,\alpha}f)^*(s), \] where * denotes, as usual, the non-increasing rearrangement of f. Several applications of this careful study are given.
Mathematics Subject Classification ID: 42B25
Mathematics Subject Classification ID: 46E30
zbMATH DE Number: 3901395
zbMATH Keywords: local maximal function
zbMATH Keywords: Peetre K-functional
zbMATH Keywords: \(L^{1,\infty }({\mathbb{R}}^ n)\)
zbMATH Keywords: \(BMO({\mathbb{R}}^ n)\)
author: Björn D. Jawerth
author: Alberto Torchinsky
reviewed by: Carlos A. Berenstein
MaRDI profile type: Publication
full work available at URL: 0021-9045(85)90102-9
OpenAlex ID: W4213353765
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description: scientific article; zbMATH DE number 3901395